In SPH, there are various formulations for viscosity. In the momentum equation, the introduction of a viscous term is necessary not only to consider viscid fluids and no slip boundary conditions, but also to provide the stability to the system and to prevent inter-particle penetration. The artificial viscosity term \(\Pi _{ij}\)  is added to pressure terms within the momentum equation (REF ). The artificial viscosity \(\Pi _{ij}\)  has the form [1]}
\(\Pi _{ij} ={\left\lbrace \begin{array}{ll}\dfrac{-\alpha c\mu _{ij}+\beta \mu _{ij}^2}{(\rho _i + \rho _j)/2}, & \text{$(\mathbf {v_{i}-\mathbf {v_j}}).(\mathbf {r_{i}-\mathbf {r_j}})<0$}\\0, & \text{$(\mathbf {v_{i}-\mathbf {v_j}}).(\mathbf {r_{i}-\mathbf {r_j}})>0$}\end{array}\right.}\) 
